On the complete objects in the category of T₀ closure spaces
by
Didier Deses
VUB, Brussels
Coauthors: E. Colebunders, E. Giuli

In [2] and [1] the concept of (sub-) firmly (epi-) reflective subcategory was introduced in order to study the 'complete' objects of a category. In the present paper we apply these results to the subcategory CLS₀ of T₀-objects of the category CLS of closure spaces and continuous maps. The objects in the resulting category of 'complete' T₀-objects will be constructed as closed subspaces of powers of the Sierpinski space w.r.t. the regular CLS₀-closure as defined in [4]. This closure coincides with the front-closure and the Zariski closure (cfr. [3]). An internal characterization of these 'complete' spaces will be given and it will be shown that they correspond to the algebraic closure spaces described in [3]. Finally we shall prove a duality between the category consisting of these 'complete' objects and the category of complete lattices with ( \/ , 1)-preserving maps. This duality can be compared to the classical one between sober topological spaces and frames.

References

1. C. C. L. Brümmer and E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolin. 33 (1), 1992, 131-147.
2. G. C. L. Brümmer, E. Giuli and H. Herrlich, Epireflections which are completions, Cahiers Topologie Géom. Différentielle Catég. 33 (1), 1992, 71-73.
3. Yves Diers, Categories of algebraic sets, Appl. Categ. Structures 4 (2-3), 1996, 329-341 (The European Colloquium of Category Theory, Tours, 1994).
4. D. Dikranjan, F. Giuli and A. Tozzi, Topological categories and closure operators, Quaestiones Math. 11 (3), 1988, 323-337.

Date received: July 17, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-41.